We now bound the variance of \(\hat{\sigma }^2_n\) . Using the variance formula for quadratic forms (cf. Theorem 11.23 in [1]}), we have
\(\mathrm {Var}(\hat{\sigma }^2_n)&=2\lambda ^2\sigma _0^4\mathrm {tr}([K(X,X)+n\lambda \mathbf {I}_n]^{-2})+4\lambda ^2 \sigma _0^2\mathbf {f}^T[K(X,X)+n\lambda \mathbf {I}_n]^{-2}\mathbf {f}\\&\le 2\lambda ^2\sigma _0^4\cdot n(n\lambda )^{-2}+4\lambda ^2 \sigma _0^2\cdot \lambda _{\max }([K(X,X)+n\lambda \mathbf {I}_n]^{-2})\Vert \mathbf {f}\Vert _2^2\\&\le 2\sigma _0^4 n^{-1}+4\sigma _0^2\Vert \mathbf {f}\Vert _2^2 n^{-2}.\) 
